There is no set threshold because -- as you posted -- Y is a r.v. By implication so is max{a,Y}. I am guessing that the cdf of max{a,Y} would be some linear combination of CDF(Y|Y>a) and the mass point Y=a (representing all the occurances of Y<a). Even after obtaining CDF(max{a,Y}) you still need to figure out the CDF of the related r.v. (X - max{a,Y}).